I'm still looking for something more detailed, but more specifically, (according to two different sources 1 2) in the 1990s Finland has already unilaterally denounced part III (art. 13-22) of the 1947 treaty, which imposed limits on the size of its armed forces and various other limitations like even not buying any civil aircraft with German components (art 21). They already had done some reinterpretation in that regard in the 1960s with respect to article 17 that had barred from having any missiles, reinterpreted then to allow them to have at least anti-aircraft missiles. (Likewise the other provisions against guided missiles and torpedo boats could be easily interpreted as being violated by the Hamina-class missile boats, which also carry some lightweight torpedoes nowadays.)
It's also worth noting that the 1948 friendship treaty with the USSR actually contained foreign policy limitations that would have been much more relevant to your question (because e.g. "Article Four prohibited Finland from ‘establishing or joining any alliances that are targeted against the other high party’"). But it was replaced in 1992 by a new treaty with Russia. In the aftermath, in 1995, Finland joined the EU, which as you probably know also has a common defense clause in article 42.7.
The NATO treaty in itself doesn't mandate any size of armed forces, or what kinds of weapons to have, so on the face of it, it adds nothing in terms of violations of the 1947 treaty. Conversely, it's usually assumed that NATO countries need to spend at least 2% of their budget on the armed forces, but the 1947 USSR-Finland treaty doesn't put any limitation on the Finnish armed forces budget- or spending-wise. (Also the 2% spending is not in the NATO treaty either, being the result of a guideline.)
So the answer in relation to that part of the 1947 treaty really depends on what kinds and quantity of weapons Finland would acquire in view of or after joining NATO, assuming that Russia still intends to hold Finland to the letter of the 1947 treaty, which isn't entirely clear to me (right now). I know they've recently warned Finland and Sweden not to join NATO, but I don't recall specific treaties [with Russia] being mentioned in that context.
I should probably add here that Italy, Hungary, Bulgaria, and Romania all have similar 1947 treaties that prohibited much of their rearmament (without UNSC approval), from specific numeric forces limits to outright prohibitions on specific weapons like submarines or guided missiles. Those countries all broke those treaties with respect to those limitations. The case of Italy is better documented in Western sources, as their violations came with the explicit blessing of the Western Allies, but these also alleged that the Soviet bloc countries on that list did the same, around the same time . (The aforementioned Soviet bloc countries also flatly refused to implement the Conciliation Commissions specified in those treaties, leading to an ICJ case.)
As far as I can tell, the 1947 treaty with Finland is silent on other issues that may be entailed by NATO membership, like e.g. the stationing of foreign (especially US or UK) forces on Finland's soil.
Ah, I actually missed something: the 1947 treaty with Finland does something the other 4 mentioned don't have an equivalent: it also restores via its article 3 the earlier 1940 peace treaty between the USSR and Finland (minus articles 4, 5 and 6 of the latter)--treaty concluded after the Winter War. But that 1940 treaty also has an article 3 (restored through the 1947 treaty) that says:
Both contracting parties undertake each to refrain from any attack upon the other and to make no alliance and to participate in no coalition directed against either of the contracting parties.
So, since Russia says NATO is an alliance directed against it, I suppose Russia could raise the issue of article 3 of that 1940 treaty as being contravened by Finland joining NATO... although I haven't heard Russia making that specific
I suppose Finland could claim that Russia also/already broke that treaty in the same way, since as the USSR it formed a similar military pact (Warsaw Pact), and more recently it did the same with the CSTO. Both of these claim to be defensive organizations just like NATO claims.